diff --git a/Lib/smtplib.py b/Lib/smtplib.py
index c949d77..562a182 100644
--- a/Lib/smtplib.py
+++ b/Lib/smtplib.py
@@ -751,7 +751,10 @@
                 esmtp_opts.append(option)
         (code, resp) = self.mail(from_addr, esmtp_opts)
         if code != 250:
-            self.rset()
+            if code == 421:
+                self.close()
+            else:
+                self.rset()
             raise SMTPSenderRefused(code, resp, from_addr)
         senderrs = {}
         if isinstance(to_addrs, str):
@@ -760,13 +763,19 @@
             (code, resp) = self.rcpt(each, rcpt_options)
             if (code != 250) and (code != 251):
                 senderrs[each] = (code, resp)
+            if code == 421:
+                self.close()
+                raise SMTPRecipientsRefused(senderrs)
         if len(senderrs) == len(to_addrs):
             # the server refused all our recipients
             self.rset()
             raise SMTPRecipientsRefused(senderrs)
         (code, resp) = self.data(msg)
         if code != 250:
-            self.rset()
+            if code == 421:
+                self.close()
+            else:
+                self.rset()
             raise SMTPDataError(code, resp)
         #if we got here then somebody got our mail
         return senderrs
diff --git a/Lib/test/test_smtplib.py b/Lib/test/test_smtplib.py
index befc49e..c0d9dbd 100644
--- a/Lib/test/test_smtplib.py
+++ b/Lib/test/test_smtplib.py
@@ -592,8 +592,12 @@
 # Simulated SMTP channel & server
 class SimSMTPChannel(smtpd.SMTPChannel):
 
-    # For testing failures in QUIT when using the context manager API.
     quit_response = None
+    mail_response = None
+    rcpt_response = None
+    data_response = None
+    rcpt_count = 0
+    rset_count = 0
 
     def __init__(self, extra_features, *args, **kw):
         self._extrafeatures = ''.join(
@@ -608,6 +612,8 @@
                 '250-DELIVERBY\r\n')
         resp = resp + self._extrafeatures + '250 HELP'
         self.push(resp)
+        self.seen_greeting = arg
+        self.extended_smtp = True
 
     def smtp_VRFY(self, arg):
         # For max compatibility smtplib should be sending the raw address.
@@ -646,30 +652,50 @@
             self.push('550 No access for you!')
 
     def smtp_QUIT(self, arg):
-        # args is ignored
         if self.quit_response is None:
             super(SimSMTPChannel, self).smtp_QUIT(arg)
         else:
             self.push(self.quit_response)
             self.close_when_done()
 
+    def smtp_MAIL(self, arg):
+        if self.mail_response is None:
+            super().smtp_MAIL(arg)
+        else:
+            self.push(self.mail_response)
+
+    def smtp_RCPT(self, arg):
+        if self.rcpt_response is None:
+            super().smtp_RCPT(arg)
+            return
+        self.push(self.rcpt_response[self.rcpt_count])
+        self.rcpt_count += 1
+
+    def smtp_RSET(self, arg):
+        super().smtp_RSET(arg)
+        self.rset_count += 1
+
+    def smtp_DATA(self, arg):
+        if self.data_response is None:
+            super().smtp_DATA(arg)
+        else:
+            self.push(self.data_response)
+
     def handle_error(self):
         raise
 
 
 class SimSMTPServer(smtpd.SMTPServer):
 
-    # For testing failures in QUIT when using the context manager API.
-    quit_response = None
+    channel_class = SimSMTPChannel
 
     def __init__(self, *args, **kw):
         self._extra_features = []
         smtpd.SMTPServer.__init__(self, *args, **kw)
 
     def handle_accepted(self, conn, addr):
-        self._SMTPchannel = SimSMTPChannel(
+        self._SMTPchannel = self.channel_class(
             self._extra_features, self, conn, addr)
-        self._SMTPchannel.quit_response = self.quit_response
 
     def process_message(self, peer, mailfrom, rcpttos, data):
         pass
@@ -809,18 +835,48 @@
         self.assertRaises(smtplib.SMTPServerDisconnected, smtp.send, b'foo')
 
     def test_with_statement_QUIT_failure(self):
-        self.serv.quit_response = '421 QUIT FAILED'
         with self.assertRaises(smtplib.SMTPResponseException) as error:
             with smtplib.SMTP(HOST, self.port) as smtp:
+                self.serv._SMTPchannel.quit_response = '421 QUIT FAILED'
                 smtp.noop()
         self.assertEqual(error.exception.smtp_code, 421)
         self.assertEqual(error.exception.smtp_error, b'QUIT FAILED')
-        # We don't need to clean up self.serv.quit_response because a new
-        # server is always instantiated in the setUp().
 
     #TODO: add tests for correct AUTH method fallback now that the
     #test infrastructure can support it.
 
+    # Issue 5713: make sure close, not rset, is called if we get a 421 error
+    def test_421_from_mail_cmd(self):
+        smtp = smtplib.SMTP(HOST, self.port, local_hostname='localhost', timeout=15)
+        self.serv._SMTPchannel.mail_response = '421 closing connection'
+        with self.assertRaises(smtplib.SMTPSenderRefused):
+            smtp.sendmail('John', 'Sally', 'test message')
+        self.assertIsNone(smtp.sock)
+        self.assertEqual(self.serv._SMTPchannel.rcpt_count, 0)
+
+    def test_421_from_rcpt_cmd(self):
+        smtp = smtplib.SMTP(HOST, self.port, local_hostname='localhost', timeout=15)
+        self.serv._SMTPchannel.rcpt_response = ['250 accepted', '421 closing']
+        with self.assertRaises(smtplib.SMTPRecipientsRefused) as r:
+            smtp.sendmail('John', ['Sally', 'Frank', 'George'], 'test message')
+        self.assertIsNone(smtp.sock)
+        self.assertEqual(self.serv._SMTPchannel.rset_count, 0)
+        self.assertDictEqual(r.exception.args[0], {'Frank': (421, b'closing')})
+
+    def test_421_from_data_cmd(self):
+        class MySimSMTPChannel(SimSMTPChannel):
+            def found_terminator(self):
+                if self.smtp_state == self.DATA:
+                    self.push('421 closing')
+                else:
+                    super().found_terminator()
+        self.serv.channel_class = MySimSMTPChannel
+        smtp = smtplib.SMTP(HOST, self.port, local_hostname='localhost', timeout=15)
+        with self.assertRaises(smtplib.SMTPDataError):
+            smtp.sendmail('John@foo.org', ['Sally@foo.org'], 'test message')
+        self.assertIsNone(smtp.sock)
+        self.assertEqual(self.serv._SMTPchannel.rcpt_count, 0)
+
 
 @support.reap_threads
 def test_main(verbose=None):
diff --git a/Misc/NEWS b/Misc/NEWS
index 1377e59..fe5afdd 100644
--- a/Misc/NEWS
+++ b/Misc/NEWS
@@ -196,6 +196,10 @@
 Library
 -------
 
+- Issue #5713: smtplib now handles 421 (closing connection) error codes when
+  sending mail by closing the socket and reporting the 421 error code via the
+  exception appropriate to the command that received the error response.
+
 - Issue #17192: Update the ctypes module's libffi to v3.0.13.  This
   specifically addresses a stack misalignment issue on x86 and issues on
   some more recent platforms.